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| {{DISPLAYTITLE:Propositions As Types}} | | {{DISPLAYTITLE:Propositions As Types}} |
− | <pre>
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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− | IDS -- PAT
| + | ==Note 1== |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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− | PAT. Propositions As Types
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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− | PAT. Note 1
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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| + | <pre> |
| Identity, or the Identifier | | Identity, or the Identifier |
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| Check. | | Check. |
| + | </pre> |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | ==Note 2== |
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− | PAT. Note 2
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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| + | <pre> |
| Identity, or the Identifier (cont.) | | Identity, or the Identifier (cont.) |
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| and to carry along the type information as we go, ending up with | | and to carry along the type information as we go, ending up with |
| a typed parse tree for I, tantamount to a proof tree for A => A. | | a typed parse tree for I, tantamount to a proof tree for A => A. |
| + | </pre> |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | ==Note 3== |
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− | PAT. Note 3
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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| + | <pre> |
| Identity, or the Identifier (cont.) | | Identity, or the Identifier (cont.) |
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| o-----------------------------------------------------------o | | o-----------------------------------------------------------o |
| + | </pre> |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | ==Note 4== |
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− | PAT. Note 4
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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| + | <pre> |
| | NB. I am working from rough notes that | | | NB. I am working from rough notes that |
| | I wrote out in the Fall of 1996, and it | | | I wrote out in the Fall of 1996, and it |
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| Therefore, the type of the application K(KS) is A=>A. | | Therefore, the type of the application K(KS) is A=>A. |
| + | </pre> |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | ==Note 5== |
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− | PAT. Note 5
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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| + | <pre> |
| Composition, or the Composer | | Composition, or the Composer |
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| P = (K((SK)S)) | | P = (K((SK)S)) |
| + | </pre> |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | ==Note 6== |
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− | PAT. Note 6
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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| + | <pre> |
| Composition, or the Composer (cont.) | | Composition, or the Composer (cont.) |
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| propositional calculus, based on the combinator axioms, | | propositional calculus, based on the combinator axioms, |
| K : A => (B=>A) and S : (A=>(B=>C)) => ((A=>B)=>(A=>C)). | | K : A => (B=>A) and S : (A=>(B=>C)) => ((A=>B)=>(A=>C)). |
| + | </pre> |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | ==Note 7== |
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− | PAT. Note 7
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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| + | <pre> |
| Composition, or the Composer (cont.) | | Composition, or the Composer (cont.) |
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| QED. | | QED. |
| + | </pre> |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | ==Note 8== |
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− | PAT. Note 8
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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| + | <pre> |
| Composition, or the Composer (cont.) | | Composition, or the Composer (cont.) |
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| o---------------------------------------------------------------------o | | o---------------------------------------------------------------------o |
| + | </pre> |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | ==Note 9== |
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− | PAT. Note 9
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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| + | <pre> |
| Composition, or the Composer (cont.) | | Composition, or the Composer (cont.) |
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| P = (K((SK)S)) : (B=>C)=>((A=>B)=>(A=>C)) | | P = (K((SK)S)) : (B=>C)=>((A=>B)=>(A=>C)) |
| + | </pre> |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | ==Note 10== |
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− | PAT. Note 10
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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| + | <pre> |
| Composition, or the Composer (concl.) | | Composition, or the Composer (concl.) |
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| amounting in effect to "Facts" being recycled as "Cases". | | amounting in effect to "Facts" being recycled as "Cases". |
| Square brackets are also used to mark the intended result. | | Square brackets are also used to mark the intended result. |
| + | </pre> |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | ==Note 11== |
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− | PAT. Note 11
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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| + | <pre> |
| Self-Documentation | | Self-Documentation |
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| o-----------------------------------------------------------o | | o-----------------------------------------------------------o |
| + | </pre> |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | ==Note 12== |
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− | PAT. Note 12
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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| + | <pre> |
| Self-Documentation, Developmental Data Structures (cont.) | | Self-Documentation, Developmental Data Structures (cont.) |
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| That's the sketch as best I can reconstruct it from my notes. | | That's the sketch as best I can reconstruct it from my notes. |
| + | </pre> |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | ==Note 13== |
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− | PAT. Note 13
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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| + | <pre> |
| Triadic Analogy: Analogy Between a Couple of Three-Place Relations | | Triadic Analogy: Analogy Between a Couple of Three-Place Relations |
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| o-------------------------------------------------o | | o-------------------------------------------------o |
| + | </pre> |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | ==Note 14== |
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− | PAT. Note 14
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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| + | <pre> |
| Transposition, or the Transposer | | Transposition, or the Transposer |
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| as a sequence of manipulations on formal identifiers, or on | | as a sequence of manipulations on formal identifiers, or on |
| symbols taken as objects in themselves. | | symbols taken as objects in themselves. |
| + | </pre> |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | ==Note 15== |
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− | PAT. Note 15
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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| + | <pre> |
| Transposition, or the Transposer (cont.) | | Transposition, or the Transposer (cont.) |
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| T = (KK)(((S((KK)S))((SK)S))S) | | T = (KK)(((S((KK)S))((SK)S))S) |
| + | </pre> |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | ==Note 16== |
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− | PAT. Note 16
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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| + | <pre> |
| Transposition, or the Transposer (cont.) | | Transposition, or the Transposer (cont.) |
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| read as a proposition, is a theorem of intuitionistic | | read as a proposition, is a theorem of intuitionistic |
| propositional calculus. | | propositional calculus. |
| + | </pre> |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | ==Note 17== |
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− | PAT. Note 17
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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| + | <pre> |
| Transposition, or the Transposer (cont.) | | Transposition, or the Transposer (cont.) |
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| o-------------------------------------------------o | | o-------------------------------------------------o |
| + | </pre> |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | ==Note 18== |
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− | PAT. Note 18
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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| + | <pre> |
| Transposition, or the Transposer (cont.) | | Transposition, or the Transposer (cont.) |
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| denote whatever it is, the supposed object, that | | denote whatever it is, the supposed object, that |
| "(F((SK)S))", the occurrent sign, denotes. | | "(F((SK)S))", the occurrent sign, denotes. |
| + | </pre> |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | ==Note 19== |
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− | PAT. Note 19
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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| + | <pre> |
| Transposition, or the Transposer (cont.) | | Transposition, or the Transposer (cont.) |
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| o-----------------------------------------------------------o | | o-----------------------------------------------------------o |
| + | </pre> |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | ==Note 20== |
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− | PAT. Note 20
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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| + | <pre> |
| Transposition, or the Transposer (cont.) | | Transposition, or the Transposer (cont.) |
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| to reconstruct the true embryology or living physiology | | to reconstruct the true embryology or living physiology |
| of discovery involved. | | of discovery involved. |
| + | </pre> |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | ==Note 21== |
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− | PAT. Note 21
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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| + | <pre> |
| Transposition, or the Transposer (cont.) | | Transposition, or the Transposer (cont.) |
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| o---------------------------------------------------------------------o | | o---------------------------------------------------------------------o |
| + | </pre> |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | ==Note 22== |
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− | PAT. Note 22
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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| + | <pre> |
| Transposition, or the Transposer (cont.) | | Transposition, or the Transposer (cont.) |
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| o---------------------------------------------------------------------o | | o---------------------------------------------------------------------o |
| + | </pre> |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | ==Note 23== |
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− | PAT. Note 23
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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| + | <pre> |
| Transposition, or the Transposer (cont.) | | Transposition, or the Transposer (cont.) |
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| o---------------------------------------------------------------------o | | o---------------------------------------------------------------------o |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | <pre> |
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− | PAT. Note 24
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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| o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
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